/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.koat /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(NON_POLY, ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.koat # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The runtime complexity of the given CpxIntTrs could be proven to be BOUNDS(INF, INF). (0) CpxIntTrs (1) Loat Proof [FINISHED, 1362 ms] (2) BOUNDS(INF, INF) ---------------------------------------- (0) Obligation: Complexity Int TRS consisting of the following rules: eval_real2_start(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb0_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_bb0_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_0(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_0(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_1(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_1(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_2(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_2(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_3(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_3(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_4(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_4(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_5(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_5(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_6(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_6(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_7(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_7(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_8(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_8(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb1_in(1, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE eval_real2_bb1_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb2_in(v_again_0, 0, v_again_2, 0, v_len)) :|: v_again_0 < 0 eval_real2_bb1_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb2_in(v_again_0, 0, v_again_2, 0, v_len)) :|: v_again_0 > 0 eval_real2_bb1_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb6_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: v_again_0 >= 0 && v_again_0 <= 0 eval_real2_bb2_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb3_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: v_i_0 < v_len - 1 eval_real2_bb2_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb1_in(v_again_1, v_again_1, v_again_2, v_i_0, v_len)) :|: v_i_0 >= v_len - 1 eval_real2_bb3_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb4_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: nondef_0 > nondef_1 eval_real2_bb3_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb5_in(v_again_0, v_again_1, v_again_1, v_i_0, v_len)) :|: nondef_0 <= nondef_1 eval_real2_bb4_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb5_in(v_again_0, v_again_1, 1, v_i_0, v_len)) :|: TRUE eval_real2_bb5_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_bb2_in(v_again_0, v_again_2, v_again_2, v_i_0 + 1, v_len)) :|: TRUE eval_real2_bb6_in(v_again_0, v_again_1, v_again_2, v_i_0, v_len) -> Com_1(eval_real2_stop(v_again_0, v_again_1, v_again_2, v_i_0, v_len)) :|: TRUE The start-symbols are:[eval_real2_start_5] ---------------------------------------- (1) Loat Proof (FINISHED) ### Pre-processing the ITS problem ### Initial linear ITS problem Start location: evalreal2start 0: evalreal2start -> evalreal2bb0in : [], cost: 1 1: evalreal2bb0in -> evalreal20 : [], cost: 1 2: evalreal20 -> evalreal21 : [], cost: 1 3: evalreal21 -> evalreal22 : [], cost: 1 4: evalreal22 -> evalreal23 : [], cost: 1 5: evalreal23 -> evalreal24 : [], cost: 1 6: evalreal24 -> evalreal25 : [], cost: 1 7: evalreal25 -> evalreal26 : [], cost: 1 8: evalreal26 -> evalreal27 : [], cost: 1 9: evalreal27 -> evalreal28 : [], cost: 1 10: evalreal28 -> evalreal2bb1in : A'=1, [], cost: 1 11: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ 0>=1+A ], cost: 1 12: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ A>=1 ], cost: 1 13: evalreal2bb1in -> evalreal2bb6in : [ A==0 ], cost: 1 14: evalreal2bb2in -> evalreal2bb3in : [ D>=2+C ], cost: 1 15: evalreal2bb2in -> evalreal2bb1in : A'=B, [ 1+C>=D ], cost: 1 16: evalreal2bb3in -> evalreal2bb4in : [ free>=1+free_1 ], cost: 1 17: evalreal2bb3in -> evalreal2bb5in : E'=B, [ free_3>=free_2 ], cost: 1 18: evalreal2bb4in -> evalreal2bb5in : E'=1, [], cost: 1 19: evalreal2bb5in -> evalreal2bb2in : B'=E, C'=1+C, [], cost: 1 20: evalreal2bb6in -> evalreal2stop : [], cost: 1 Checking for constant complexity: The following rule is satisfiable with cost >= 1, yielding constant complexity: 0: evalreal2start -> evalreal2bb0in : [], cost: 1 Removed unreachable and leaf rules: Start location: evalreal2start 0: evalreal2start -> evalreal2bb0in : [], cost: 1 1: evalreal2bb0in -> evalreal20 : [], cost: 1 2: evalreal20 -> evalreal21 : [], cost: 1 3: evalreal21 -> evalreal22 : [], cost: 1 4: evalreal22 -> evalreal23 : [], cost: 1 5: evalreal23 -> evalreal24 : [], cost: 1 6: evalreal24 -> evalreal25 : [], cost: 1 7: evalreal25 -> evalreal26 : [], cost: 1 8: evalreal26 -> evalreal27 : [], cost: 1 9: evalreal27 -> evalreal28 : [], cost: 1 10: evalreal28 -> evalreal2bb1in : A'=1, [], cost: 1 11: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ 0>=1+A ], cost: 1 12: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ A>=1 ], cost: 1 14: evalreal2bb2in -> evalreal2bb3in : [ D>=2+C ], cost: 1 15: evalreal2bb2in -> evalreal2bb1in : A'=B, [ 1+C>=D ], cost: 1 16: evalreal2bb3in -> evalreal2bb4in : [ free>=1+free_1 ], cost: 1 17: evalreal2bb3in -> evalreal2bb5in : E'=B, [ free_3>=free_2 ], cost: 1 18: evalreal2bb4in -> evalreal2bb5in : E'=1, [], cost: 1 19: evalreal2bb5in -> evalreal2bb2in : B'=E, C'=1+C, [], cost: 1 Simplified all rules, resulting in: Start location: evalreal2start 0: evalreal2start -> evalreal2bb0in : [], cost: 1 1: evalreal2bb0in -> evalreal20 : [], cost: 1 2: evalreal20 -> evalreal21 : [], cost: 1 3: evalreal21 -> evalreal22 : [], cost: 1 4: evalreal22 -> evalreal23 : [], cost: 1 5: evalreal23 -> evalreal24 : [], cost: 1 6: evalreal24 -> evalreal25 : [], cost: 1 7: evalreal25 -> evalreal26 : [], cost: 1 8: evalreal26 -> evalreal27 : [], cost: 1 9: evalreal27 -> evalreal28 : [], cost: 1 10: evalreal28 -> evalreal2bb1in : A'=1, [], cost: 1 11: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ 0>=1+A ], cost: 1 12: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ A>=1 ], cost: 1 14: evalreal2bb2in -> evalreal2bb3in : [ D>=2+C ], cost: 1 15: evalreal2bb2in -> evalreal2bb1in : A'=B, [ 1+C>=D ], cost: 1 16: evalreal2bb3in -> evalreal2bb4in : [], cost: 1 17: evalreal2bb3in -> evalreal2bb5in : E'=B, [], cost: 1 18: evalreal2bb4in -> evalreal2bb5in : E'=1, [], cost: 1 19: evalreal2bb5in -> evalreal2bb2in : B'=E, C'=1+C, [], cost: 1 ### Simplification by acceleration and chaining ### Eliminated locations (on linear paths): Start location: evalreal2start 30: evalreal2start -> evalreal2bb1in : A'=1, [], cost: 11 11: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ 0>=1+A ], cost: 1 12: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ A>=1 ], cost: 1 14: evalreal2bb2in -> evalreal2bb3in : [ D>=2+C ], cost: 1 15: evalreal2bb2in -> evalreal2bb1in : A'=B, [ 1+C>=D ], cost: 1 17: evalreal2bb3in -> evalreal2bb5in : E'=B, [], cost: 1 31: evalreal2bb3in -> evalreal2bb5in : E'=1, [], cost: 2 19: evalreal2bb5in -> evalreal2bb2in : B'=E, C'=1+C, [], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalreal2start 30: evalreal2start -> evalreal2bb1in : A'=1, [], cost: 11 11: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ 0>=1+A ], cost: 1 12: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ A>=1 ], cost: 1 15: evalreal2bb2in -> evalreal2bb1in : A'=B, [ 1+C>=D ], cost: 1 32: evalreal2bb2in -> evalreal2bb5in : E'=B, [ D>=2+C ], cost: 2 33: evalreal2bb2in -> evalreal2bb5in : E'=1, [ D>=2+C ], cost: 3 19: evalreal2bb5in -> evalreal2bb2in : B'=E, C'=1+C, [], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalreal2start 30: evalreal2start -> evalreal2bb1in : A'=1, [], cost: 11 11: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ 0>=1+A ], cost: 1 12: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ A>=1 ], cost: 1 15: evalreal2bb2in -> evalreal2bb1in : A'=B, [ 1+C>=D ], cost: 1 34: evalreal2bb2in -> evalreal2bb2in : B'=B, C'=1+C, E'=B, [ D>=2+C ], cost: 3 35: evalreal2bb2in -> evalreal2bb2in : B'=1, C'=1+C, E'=1, [ D>=2+C ], cost: 4 Accelerating simple loops of location 12. Simplified some of the simple loops (and removed duplicate rules). Accelerating the following rules: 34: evalreal2bb2in -> evalreal2bb2in : C'=1+C, E'=B, [ D>=2+C ], cost: 3 35: evalreal2bb2in -> evalreal2bb2in : B'=1, C'=1+C, E'=1, [ D>=2+C ], cost: 4 Accelerated rule 34 with metering function -1-C+D, yielding the new rule 36. Accelerated rule 35 with metering function -1-C+D, yielding the new rule 37. Removing the simple loops: 34 35. Accelerated all simple loops using metering functions (where possible): Start location: evalreal2start 30: evalreal2start -> evalreal2bb1in : A'=1, [], cost: 11 11: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ 0>=1+A ], cost: 1 12: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ A>=1 ], cost: 1 15: evalreal2bb2in -> evalreal2bb1in : A'=B, [ 1+C>=D ], cost: 1 36: evalreal2bb2in -> evalreal2bb2in : C'=-1+D, E'=B, [ D>=2+C ], cost: -3-3*C+3*D 37: evalreal2bb2in -> evalreal2bb2in : B'=1, C'=-1+D, E'=1, [ D>=2+C ], cost: -4-4*C+4*D Chained accelerated rules (with incoming rules): Start location: evalreal2start 30: evalreal2start -> evalreal2bb1in : A'=1, [], cost: 11 11: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ 0>=1+A ], cost: 1 12: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=0, [ A>=1 ], cost: 1 38: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=-1+D, E'=0, [ 0>=1+A && D>=2 ], cost: -2+3*D 39: evalreal2bb1in -> evalreal2bb2in : B'=0, C'=-1+D, E'=0, [ A>=1 && D>=2 ], cost: -2+3*D 40: evalreal2bb1in -> evalreal2bb2in : B'=1, C'=-1+D, E'=1, [ 0>=1+A && D>=2 ], cost: -3+4*D 41: evalreal2bb1in -> evalreal2bb2in : B'=1, C'=-1+D, E'=1, [ A>=1 && D>=2 ], cost: -3+4*D 15: evalreal2bb2in -> evalreal2bb1in : A'=B, [ 1+C>=D ], cost: 1 Eliminated locations (on tree-shaped paths): Start location: evalreal2start 30: evalreal2start -> evalreal2bb1in : A'=1, [], cost: 11 42: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=0, [ 0>=1+A && 1>=D ], cost: 2 43: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=0, [ A>=1 && 1>=D ], cost: 2 44: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ 0>=1+A && D>=2 ], cost: -1+3*D 45: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ A>=1 && D>=2 ], cost: -1+3*D 46: evalreal2bb1in -> evalreal2bb1in : A'=1, B'=1, C'=-1+D, E'=1, [ 0>=1+A && D>=2 ], cost: -2+4*D 47: evalreal2bb1in -> evalreal2bb1in : A'=1, B'=1, C'=-1+D, E'=1, [ A>=1 && D>=2 ], cost: -2+4*D Applied pruning (of leafs and parallel rules): Start location: evalreal2start 30: evalreal2start -> evalreal2bb1in : A'=1, [], cost: 11 42: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=0, [ 0>=1+A && 1>=D ], cost: 2 44: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ 0>=1+A && D>=2 ], cost: -1+3*D 45: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ A>=1 && D>=2 ], cost: -1+3*D 46: evalreal2bb1in -> evalreal2bb1in : A'=1, B'=1, C'=-1+D, E'=1, [ 0>=1+A && D>=2 ], cost: -2+4*D 47: evalreal2bb1in -> evalreal2bb1in : A'=1, B'=1, C'=-1+D, E'=1, [ A>=1 && D>=2 ], cost: -2+4*D Accelerating simple loops of location 11. Accelerating the following rules: 42: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=0, [ 0>=1+A && 1>=D ], cost: 2 44: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ 0>=1+A && D>=2 ], cost: -1+3*D 45: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ A>=1 && D>=2 ], cost: -1+3*D 46: evalreal2bb1in -> evalreal2bb1in : A'=1, B'=1, C'=-1+D, E'=1, [ 0>=1+A && D>=2 ], cost: -2+4*D 47: evalreal2bb1in -> evalreal2bb1in : A'=1, B'=1, C'=-1+D, E'=1, [ A>=1 && D>=2 ], cost: -2+4*D Found no metering function for rule 42. Found no metering function for rule 44. Found no metering function for rule 45. Found no metering function for rule 46. Accelerated rule 47 with NONTERM, yielding the new rule 48. Removing the simple loops: 47. Accelerated all simple loops using metering functions (where possible): Start location: evalreal2start 30: evalreal2start -> evalreal2bb1in : A'=1, [], cost: 11 42: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=0, [ 0>=1+A && 1>=D ], cost: 2 44: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ 0>=1+A && D>=2 ], cost: -1+3*D 45: evalreal2bb1in -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ A>=1 && D>=2 ], cost: -1+3*D 46: evalreal2bb1in -> evalreal2bb1in : A'=1, B'=1, C'=-1+D, E'=1, [ 0>=1+A && D>=2 ], cost: -2+4*D 48: evalreal2bb1in -> [19] : [ A>=1 && D>=2 && -2+4*D>=1 ], cost: NONTERM Chained accelerated rules (with incoming rules): Start location: evalreal2start 30: evalreal2start -> evalreal2bb1in : A'=1, [], cost: 11 49: evalreal2start -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ D>=2 ], cost: 10+3*D 50: evalreal2start -> [19] : A'=1, [ D>=2 && -2+4*D>=1 ], cost: NONTERM Removed unreachable locations (and leaf rules with constant cost): Start location: evalreal2start 49: evalreal2start -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ D>=2 ], cost: 10+3*D 50: evalreal2start -> [19] : A'=1, [ D>=2 && -2+4*D>=1 ], cost: NONTERM ### Computing asymptotic complexity ### Fully simplified ITS problem Start location: evalreal2start 49: evalreal2start -> evalreal2bb1in : A'=0, B'=0, C'=-1+D, E'=0, [ D>=2 ], cost: 10+3*D 50: evalreal2start -> [19] : A'=1, [ D>=2 && -2+4*D>=1 ], cost: NONTERM Computing asymptotic complexity for rule 49 Solved the limit problem by the following transformations: Created initial limit problem: -1+D (+/+!), 10+3*D (+) [not solved] removing all constraints (solved by SMT) resulting limit problem: [solved] applying transformation rule (C) using substitution {D==n} resulting limit problem: [solved] Solution: D / n Resulting cost 10+3*n has complexity: Poly(n^1) Found new complexity Poly(n^1). Computing asymptotic complexity for rule 50 Guard is satisfiable, yielding nontermination Resulting cost NONTERM has complexity: Nonterm Found new complexity Nonterm. Obtained the following overall complexity (w.r.t. the length of the input n): Complexity: Nonterm Cpx degree: Nonterm Solved cost: NONTERM Rule cost: NONTERM Rule guard: [ D>=2 ] NO ---------------------------------------- (2) BOUNDS(INF, INF)